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Derivatives in Higher Dimensions

RolandIRL · 2010-11-28 14:57:11

say two functions f(x,y) and g(x) are differentiable at a =(5,6) for example and f(a) respectively for f:R^2 -> R and g:R->R Using the chain rule and jacobian matrices to get the derivative composition, i've ended up with a 1x2 matrix. however, is the derivative at a, the product of their jacobians or the product times…

28-11-2010 2:57pm

#1

RolandIRL

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say two functions f(x,y) and g(x) are differentiable at a =(5,6) for example and f(a) respectively for f:R^2 -> R and g:R->R
Using the chain rule and jacobian matrices to get the derivative composition, i've ended up with a 1x2 matrix. however, is the derivative at a, the product of their jacobians or the product times some vector h with ||h|| -> 0

eg (2 5) = the product of their jacobians and h =(h,k)

is the derivative (2 5) or 2h + 5k (multiplying out the vector by the jacobian product)?

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#2 29-11-2010 1:25am

Fringe

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The chain rule gives g'(f(a))f'(a). g'(f(a)) and f'(a) are the corresponding Jacobians. Multiply them together and you get the derivative. You can think of h as the direction when you look at the directional derivative.

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